To verify that the potential \(w_Q\)  has the right period to be mirror to \((X_P,D_P)\) , we can restrict to the cluster torus \(\mathbb {T}_{12}\subset U\)  to get a Laurent polynomial \(w_0 := w_Q|_{\mathbb {T}_{12}}\in \mathbb {C}[x_1^{\pm 1},x_2^{\pm 1}]\) . The only difference between the Landau–Ginzburg model \((U,w_Q)\)  and the toric Landau–Ginzburg model \((\mathbb {T}_{12},w_0)\)  is that we have extended the domain of definition of \(w_0\)  along the two exceptional curves of \(U\setminus \mathbb {T}_{12}\) . It is now just a case of verifying that \(w_0\)  is mutation equivalent to a known Laurent polynomial mirror to \(X_Q\)  (see [1]}).
